let L2 be Lattice; :: thesis: for 0L being lower-bounded Lattice

for f being Homomorphism of 0L,L2 st f is onto holds

( L2 is lower-bounded & f preserves_bottom )

let 0L be lower-bounded Lattice; :: thesis: for f being Homomorphism of 0L,L2 st f is onto holds

( L2 is lower-bounded & f preserves_bottom )

let f be Homomorphism of 0L,L2; :: thesis: ( f is onto implies ( L2 is lower-bounded & f preserves_bottom ) )

set r = f . (Bottom 0L);

assume A1: f is onto ; :: thesis: ( L2 is lower-bounded & f preserves_bottom )

then Bottom L2 = f . (Bottom 0L) by A2, LATTICES:def 16;

hence f preserves_bottom ; :: thesis: verum

for f being Homomorphism of 0L,L2 st f is onto holds

( L2 is lower-bounded & f preserves_bottom )

let 0L be lower-bounded Lattice; :: thesis: for f being Homomorphism of 0L,L2 st f is onto holds

( L2 is lower-bounded & f preserves_bottom )

let f be Homomorphism of 0L,L2; :: thesis: ( f is onto implies ( L2 is lower-bounded & f preserves_bottom ) )

set r = f . (Bottom 0L);

assume A1: f is onto ; :: thesis: ( L2 is lower-bounded & f preserves_bottom )

A2: now :: thesis: for a2 being Element of L2 holds

( (f . (Bottom 0L)) "/\" a2 = f . (Bottom 0L) & a2 "/\" (f . (Bottom 0L)) = f . (Bottom 0L) )

thus
L2 is lower-bounded
by A2; :: thesis: f preserves_bottom ( (f . (Bottom 0L)) "/\" a2 = f . (Bottom 0L) & a2 "/\" (f . (Bottom 0L)) = f . (Bottom 0L) )

let a2 be Element of L2; :: thesis: ( (f . (Bottom 0L)) "/\" a2 = f . (Bottom 0L) & a2 "/\" (f . (Bottom 0L)) = f . (Bottom 0L) )

consider a1 being Element of 0L such that

A3: f . a1 = a2 by A1, Th6;

thus (f . (Bottom 0L)) "/\" a2 = f . ((Bottom 0L) "/\" a1) by A3, D2

.= f . (Bottom 0L) ; :: thesis: a2 "/\" (f . (Bottom 0L)) = f . (Bottom 0L)

hence a2 "/\" (f . (Bottom 0L)) = f . (Bottom 0L) ; :: thesis: verum

end;consider a1 being Element of 0L such that

A3: f . a1 = a2 by A1, Th6;

thus (f . (Bottom 0L)) "/\" a2 = f . ((Bottom 0L) "/\" a1) by A3, D2

.= f . (Bottom 0L) ; :: thesis: a2 "/\" (f . (Bottom 0L)) = f . (Bottom 0L)

hence a2 "/\" (f . (Bottom 0L)) = f . (Bottom 0L) ; :: thesis: verum

then Bottom L2 = f . (Bottom 0L) by A2, LATTICES:def 16;

hence f preserves_bottom ; :: thesis: verum